# What is the difference b/w analytic, algebraic, differential geometry?

## Main Question or Discussion Point

Also, is there a special term for the geometry typically taught in high school?

And how does topology fit in here? It seems that topology is a form of geometry as well.

phinds
Gold Member
2019 Award
Geometry taught in high school is "plain geometry" and a follow-on course (or a section late in the plain geometry course) would be "spherical geometry"

Analytic and algebraic geometry are the same thing (or at least that's how the words were used 50+ years ago when I was in high school).

Those are high school topics. Differential geometry and topology are much more advanced.

You really should learn how to use Google. It will answer such questions for you pretty readily. For example, Google "what is differential geometry"

Analytic geometry either refers to high school geometry where you used coordinates, or refers to something much more advanced where you study analytic spaces and analytic/smooth maps between them.

Differential geometry is a bit more restrictive than analytic geometry and discusses mainly smooth structures without singularities and smooth maps.

Algebraic geometry refers to the geometries created by polynomial functions such as conic sections.

Also, is there a special term for the geometry typically taught in high school?
That would be Euclidean geometry

And how does topology fit in here? It seems that topology is a form of geometry as well.
Yes, topology is also a kind of geometry. To see where things fit in: http://en.wikipedia.org/wiki/Erlangen_program

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HallsofIvy
Homework Helper
Geometry taught in high school is "plain geometry" and a follow-on course (or a section late in the plain geometry course) would be "spherical geometry"
I believe the usual title is plane geometry, though admittedly it is pretty "plain"!

Analytic and algebraic geometry are the same thing (or at least that's how the words were used 50+ years ago when I was in high school).
No. Algebraic geometry is a completely different course- and considerably more advanced.
From Wikipedia "Algebraic geometry is a branch of mathematics, classically studying zeros of polynomial equations. Modern algebraic geometry is based on more abstract techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry." Definitely not high school material!

Those are high school topics. Differential geometry and topology are much more advanced.

You really should learn how to use Google. It will answer such questions for you pretty readily. For example, Google "what is differential geometry"

phinds
Gold Member
2019 Award
I believe the usual title is plane geometry, though admittedly it is pretty "plain"!
AAAAARRRRRGGGGHHHHH !!! I HATE it when I do that

disregardthat
I believe the usual title is plane geometry, though admittedly it is pretty "plain"!
I'm amazed that you say that with that quote of yours!

SteamKing
Staff Emeritus
Homework Helper
They used to teach solid geometry, which was the study of various 3-D objects, but that apparently was too tough for students to master. Plane trigonometry is a special study of triangles which relies on Euclidean geometry for important proofs. Spherical trigonometry is also another subject which has apparently fallen by the wayside, once being essential for subjects like geodesy and navigation.

mathwonk
Homework Helper
The difference is in which maps are admitted. In increasing order of specialization (and in modern advanced, not elementary high school terminology), topology is the geometry where maps are only required to be continuous, differential geometry allows only maps which are "smooth" (usually C^infinity), analytic geometry allows only maps defined locally by convergent power series, and algebraic geometry is the most restrictive of all in allowing only maps which are locally defined by polynomials or sometimes rational functions.

In the same way the objects studied in each geometry are often taken to be locally the zero locus of functions of the appropriate kind, at least in the last three geometries: a differentiable manifold is locally the zero locus of a smooth function (with maximal rank derivative), an analytic variety is locally the zero locus of a power series; an algebraic variety is locally the zero locus of a polynomial map.

Isomorphisms between two objects in each category are defined in the analogous ways: existence of mutually inverse maps of the appropriate kind.

Since the categories above require increasingly more special maps, algebraically isomorphic objects are also analytically isomorphic, differentiably isomorphic and homeomorhic. Analytically isomorphic objects are also differentiably isomorphic and homeomorphic but not necessarily algebraically isomorphic.

It is of interest to aks when one can reverse these implications, i.e. to know when one can reduce the question of isomorphism to a more general, less restrictive, category.

A big theorem implies that in the case of projective algebraic varieties analytic isomorphism does imply algebraic isomorphism. Other big theorems imply that certain differentiably isomorphic algebraic varieties have certain algebraic invariants also equal. In dimension two at least, topologically equivalent, i.e. homeomorphic surfaces are also differentiably isomorphic.

The "index" versions of the Riemann Roch theorems say that certain analytic invariants of algebraic varieties, such as the "euler characteristic" (alternating sum of dimensions of algebraic or analytic sheaf cohomology groups) of certain line bundles or vector bundles, is actually a topological invariant. E.g. for a line bundle L on a Riemann surface, the difference h^0(L) -h^1(L) of the ranks of the algebraic sheaf cohomology groups equals d + 1-g, where d is the degree and g is the topological genus, a purely topological invariant of L.

generalizations of these concepts also exist, but these are the most basic meanings of these words.

the geometry taught in high school, or what used to be taught, is Euclidean plane geometry, a special case of differential geometry. In differential geometry one studies notions involving measure and curvature, and a special case is the study of surfaces whose curvature is constant everywhere. Euclidean geometry is that most special case where the curvature is not only constant everywhere but equal to zero everywhere. I.e. high school geometry is the geometry of a flat 2 dimensional plane.

So called classical non - Euclidean geometry is usually the study of surfaces with curvature constantly equal to some negative number everywhere. Spherical geometry is the study of a surface of constant positive curvature.

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1 person
HallsofIvy
Homework Helper
I believe the usual title is plane geometry, though admittedly it is pretty "plain"!
I'm amazed that you say that with that quote of yours!
Euclid dealt with a lot more than plane geometry.

mathwonk
Homework Helper
Halls is of course right. Read Euclid for plane geometry, (geometric) algebra as far as quadratic equations, solid geometry, "Dedekind cuts", number theory, limits,....

mathwonk
Homework Helper
here is a book on analytic geometry, especially chapters 3,5:

https://www.amazon.com/dp/0821821652/?tag=pfamazon01-20

and another one, esp. chapter 3:

https://www.amazon.com/dp/0387953957/?tag=pfamazon01-20

here is a book on differential geometry:

https://www.amazon.com/dp/0132125897/?tag=pfamazon01-20

and one that combines differential and topological examples:

https://www.amazon.com/dp/0817638407/?tag=pfamazon01-20
here is a book on algebraic geometry:

https://www.amazon.com/dp/0387977163/?tag=pfamazon01-20

and a free one:

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

and here is a book on geometric topology:

https://www.amazon.com/dp/146129908X/?tag=pfamazon01-20

and another maybe better beginning one, especially chapter one:

https://www.amazon.com/dp/0387902716/?tag=pfamazon01-20

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mathwonk
Homework Helper
swampwiz: did my explanation help at all?

1 person
mathwonk
Homework Helper
trying again to be more elementary: the geometries you mention differ roughly in how many derivatives are required for the functions used in them.

in topology no derivatives are required, just continuity.

in differential geometry at least two derivatives are needed and often infinitely many are used.

in analytic geometry not only are infinitely many derivatives required, but it is also required that the taylor series they define actually represents the function locally everywhere, i.e. an analytic function must be determined just by the values of all its derivatives at one point.

in algebraic geometry, not only are infinitely many derivatives required, but it also required that for some finite n >0, all derivatives of order > n are identically zero.

swampwiz: did my explanation help at all?
It helped a bit.

The reason I asked this question is because I just downloaded a huge torrent of math textbooks and I wanted to categorize the books that deal with geometry. The torrents had some books in folders named "algebraic" and "differential". Both of these seem to be advanced topics that either require the prerequisite knowledge of abstract algebra and/or advanced calculus or analysis (the difference between those two domains is another question!) To me, geometry that only requires regular calculus (i.e., here "regular" means up through multivariate calculus, including vector calculus) is at a more basic level than that which requires those advanced prerequisites.

And just looking at an "introduction to topology" book, it seemed to require advanced calculus as well, with terminology that seemed to exceed multivariate calculus. Is it possible to understand topology decently with only multivariate calculus & differential equations (i.e., the extent of my mathematical knowledge)?

WWGD
Gold Member
2019 Award
trying again to be more elementary: the geometries you mention differ roughly in how many derivatives are required for the functions used in them.

in topology no derivatives are required, just continuity.

in differential geometry at least two derivatives are needed and often infinitely many are used.

in analytic geometry not only are infinitely many derivatives required, but it is also required that the taylor series they define actually represents the function locally everywhere, i.e. an analytic function must be determined just by the values of all its derivatives at one point.

in algebraic geometry, not only are infinitely many derivatives required, but it also required that for some finite n >0, all derivatives of order > n are identically zero.
is this related to the Erlanger program http://en.wikipedia.org/wiki/Erlangen_program ?

mathwonk